An effective method of investigation of positive maps on the set of positive definite operators

Abstract

Let A 1 be the algebra of linear operators on the n-dimensional Hilbert space H 1, and let A 2 be the algebra of linear operators of the m-dimensional Hilbert space H 2. Let L ( A 1, A 2) denote the complex space of linear maps from A 1 to A 2. By a positive map we mean an element of the space L ( A 1, A 2) (superoperator with respect to H 1) which maps positive definite operators in A 1 into positive definite operators in A 2. The aim of this paper is to present an effective method which allows to verify whether a given superoperator Λ∈ L ( A 1, A 2) is a positive map. Besides that necessary and sufficient conditions for the positive definiteness of even-degree forms in many variables are given.